For a multiday backpacking trip that feels like a vacation, East Basin is a great destination. A reasonable trip would be two days in and two days out, though it's not inconceivable for an athletic hiker to make the whole return trip in one day. However, with so many nice lakes on the way, I'd recommend adding a day or two to enjoy the area.
Day 1 - From the Lake Fork trailhead, head southeast 0.1 mile to a gate and a sign. Take the Brown Duck Lake trail southeast. After a half-mile, the trail bends northeast  the general direction of the rest of the hike. At 4 miles, a gate leads to a meadow. A stream makes a nice lunch stop. Brown Duck Lake is at 7 miles, just west of a sign indicating trails to Tworoose Pass and East Basin. You can camp here or continue to Atwine Lake about 2 miles north on the East Basin Trail. Atwine is more scenic and less crowded. Brown Duck had better fishing and easier access to more lakes. Wherever you camp, Brown Duck Basin is a nice day to spend an extra day.
Day 2 - Head north to East Basin. The first major landmark is Clements Lake, 1.25 miles from Atwine Lake. The trail turns east and winds north again 1 mile through alpine meadows and forest to East Basin Pass. Enjoy the views as you descend into the basin. At the bottom of the scree are some raspberry patches that should have ripe fruit right now and a pretty pond. Continue 1.7 miles northeast to the next significant lake. The soft meadows around it make for comfortable camping. I ended here because a hailstorm arrived.
Return the way you came.

Getting there: From Duchesne, go north on State Road 87 for 15.5 miles. As the road bends, turn north onto 21000 West. The road jogs east at 3.5 miles and becomes 20780 West. Stay on this road for just over 14 miles. The trailhead is in a large parking area just south of the Moon Lake campground.